How to Do It

Complex systems are those with many strongly interdependent variables. This
excludes systems with only a few effective variables, the kind we meet in
elementary dynamics. It also excludes systems with many independent variables;
we learn how to deal with them in elementary statistical mechanics. Complexity
appears where coupling is important, but doesn't freeze out most degrees of
freedom.

Complex systems have been infiltrating physics for about two decades, just
they have crept into biology, computer science and economics. It comes to us from two
sources. One is dynamics, studying the qualitative and effectively-stochastic
("chaotic") properties of nonlinear systems. The other is the statistical
mechanics of critical phenomena. Criticality taught us to derive large-scale
order from local interactions, and to expect that toy models of those
interactions would yield universally valid results. (The latter source is
under-appreciated by non-physicists, because "fixed point of the
renormalization group" sounds duller than "strange attractor".) Both sources
lead us to see computer simulation as an important source of knowledge.

The book has three parts: two introductory chapters, three on "mean-field
type models" and three on "agent-based models". Every chapter contains a truly
remarkable number of examples, mixing classical and up-to-date models from many
disciplines with physicists' contributions; a number of these, like the
"susceptible-infected-recovered" model from epidemiology, recur across the
chapters, revealing new aspects. The corner-stone is chapter two, where
Boccara gives a marvelous demonstration of how one models complex
systems, guiding the reader through the construction of increasingly elaborate
models of predator-prey oscillations in ecology, incorporating more and more
sophisticated mechanisms of interaction. and testing the results against
(stylized) empirical facts. Much of the rest of the book unfolds the ideas
glimpsed in this chapter.

The part on "mean-field type models" covers finite-dimensional global
dynamics. Boccara carefully states rigorous results of nonlinear dynamics,
especially bifurcation theory, together with precise definitions of the
concepts involved. He omits proofs, instead using the theorems to analyze real
models. The final part, on "agent-based models", delivers excellent,
up-to-date chapters on cellular automata (Boccara's specialty), power-law
distributions, and networks. (Readers should be aware that Boccara and
computer scientists or economists define "agent-based model" in different,
though ultimately equivalent, ways.) The chapter on power laws deserves special
mention: it gives cautionary examples showing how easy it is to mistake other
things (e.g., log-normals) for power-laws, and a prophylactic introduction to
statistical hypothesis testing. The cover promises deep thoughts on the nature
of complexity, emergence, and so forth. In fact, Boccara wisely leaves these
alone: while interesting scientific work has been done on these topics, it is
quite advanced, and obscured by a much larger volume of fluff. Instead, he
provides a guide to actual model-building, covering an astonishing range of
subjects along the way.

That said, and allowing for the need to be selective, I was struck by three
omissions. (1) What one might call "traditional physics" complex systems: spin
glasses, turbulence, polymers, and generally soft or far-from-equilibrium condensed matter. (2) The theory of
information and computation. These are crucial components in meaningful
complexity measures, but surprisingly little-studied by physicists. (Badii and
Politi's Complexity: Hierarchical
Structures and Scaling in Physics is a great, but more demanding,
guide to these subjects.) (3) Adaptation, whether at the individual and at the
population or evolutionary level. No even cases where physical methods have
been useful, like neural network learning, make it in. (Here the best
introduction I know of is G. W. Flake's The Computational Beauty of
Nature, which is at a lower mathematical level.) These topics,
however, would have needed another hundred pages at least.

The ideal audience for this book is first- or second- year physics graduate
students who have had a one-semester course in modern statistical mechanics,
and so some grasp of the ideas of criticality, fluctuations, correlation
functions, etc. Anyone so prepared will find the book clear and well worth
reading. This ought to be the standard introduction to the physics of complex
systems for the foreseeable future, and everyone seriously concerned with the
subject should read it.

Disclaimer: I got a review copy of this book from Physics
Today, but I have no stake in the book's success.